Logic puzzles can teach reasoning in a fun way that doesn't feel like work.
- Logician Raymond Smullyan devised tons of logic puzzles, but one was declared by another philosopher to be the hardest of all time.
- The problem, also known as the Three Gods Problem, is solvable, even if it doesn't seem to be.
- It depends on using complex questions to assure that any answer given is useful.
The Three Gods Problem<iframe width="730" height="430" src="https://www.youtube.com/embed/UyOGZk7WbIk" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><p> One of the more popular wordings of the problem, which MIT logic professor George Boolos <a href="https://www.readersdigest.ca/culture/hardest-logic-puzzle-ever/" target="_blank">said</a> was the hardest ever, is:<br> <br> "Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for <em>yes</em> and <em>no</em> are <em>da</em> and <em>ja</em>, in some order. You do not know which word means which."<br> <br> Boolos adds that you are allowed to ask a particular god more than one question and that Random switches between answering as if they are a truth-teller or a liar, not merely between answering "da" and "ja." <br> <br> Give yourself a minute to ponder this; we'll look at a few answers below. Ready? Okay. <strong><br> <br></strong>George Boolos' <a href="https://www.pdcnet.org/8525737F00588A37/file/31B21D0580E8B125852577CA0060ABC9/$FILE/harvardreview_1996_0006_0001_0060_0063.pdf" target="_blank" rel="noopener noreferrer">solution</a> focuses on finding either True or False through complex questions. </p><p> In logic, there is a commonly used function often written as "iff," which means "if, and only if." It would be used to say something like "The sky is blue if and only if Des Moines is in Iowa." It is a powerful tool, as it gives a true statement only when both of its components are true or both are false. If one is true and the other is false, you have a false statement. </p><p> So, if you make a statement such as "the moon is made of Gorgonzola if, and only if, Rome is in Russia," then you have made a true statement, as both parts of it are false. The statement "The moon has no air if, and only if, Rome is in Italy," is also true, as both parts of it are true. However, "The moon is made of Gorgonzola if, and only if, Albany is the capitol of New York," is false, because one of the parts of that statement is true, and the other part is not (The fact that these items don't rely on each other is immaterial for now).</p><p> In this puzzle, iff can be used here to control for the unknown value of "da" and "ja." As the answers we get can be compared with what we know they would be if the parts of our question are all true, all false, or if they differ. </p><p> Boolos would have us begin by asking god A, "Does "da" mean yes if and only if you are True if and only if B is Random?" No matter what A says, the answer you get is extremely useful. As he explains: <br> </p><p> "If A is True or False and you get the answer da, then as we have seen, B is Random, and therefore C is either True or False; but if A is True or False and you get the answer ja, then B is not Random, therefore B is either True or False… if A is Random and you get the answer da, C is not Random (neither is B, but that's irrelevant), and therefore C is either True or False; and if A is Random...and you get the answer ja, B is not random (neither is C, irrelevantly), and therefore B is either True or False."<br> <br> No matter which god A is, an answer of "da" assures that C isn't Random, and a response of "ja" means the same for B. </p><p> From here, it is a simple matter of asking whichever one you know isn't Random questions to determine if they are telling the truth, and then one on who the last god is. Boolos suggests starting with "Does da mean yes if, and only if, Rome is in Italy?" Since one part of this is accurate, we know that True will say "da," and False will say "ja," if faced with this question. </p><p> After that, you can ask the same god something like, "Does da mean yes if, and only if, A is Random?" and know exactly who is who by how they answer and the process of elimination. </p><p> If you're confused about how this works, try going over it again slowly. Remember that the essential parts are knowing what the answer will be if two positives or two negatives always come out as a positive and that two of the gods can be relied on to act consistently. </p><p> Smullyan wrote several books with other logic puzzles in them. If you liked this one and would like to learn more about the philosophical issues they investigate, or perhaps if you'd like to try a few that are a little easier to solve, you should consider reading them. A few of his puzzles can be found with explanations in this <a href="https://www.nytimes.com/interactive/2017/02/11/obituaries/smullyan-logic-puzzles.html" target="_blank" rel="noopener noreferrer">interactive</a>. </p>
Do you ever act irrationally? You probably have. Let's take a look at how to fix that.
Most of us like to suppose that we are rational people, going about our days with at least some attempt at using logic and reason. However, logical fallacies and simple mistakes are everywhere. Some wrong ways of thinking are so familiar or so easy to overlook that it is possible you're unaware that there is even a mistake being made.
Author, speaker, and public intellectual Richard Dawkins is a first-class debater on subjects as grand and reaching as the very existence (or lack thereof) of a master creator. But he's got a simple yet highly effective technique to win people over to see his point of view. Find out what it is right here.
Many people would like to have a one-on-one argument with renowned professor, author, and all-around big thinker Richard Dawkins. He's most one of the world's most prominent public intellectuals and has written over a dozen books on matters as wide-ranging as atheism and science. Because he attacks such deeply held beliefs, many people disagree with him. But how is he so effective at what he does? Simple. He imagines his argument from the other side's perspective. That way, Richard Dawkins posits, there's a much higher chance that he can land his point. Richard Dawkins' new book is Science in the Soul: Selected Writings of a Passionate Rationalist.
Is creativity a wild and free state of mind, or is it actually a pattern that others just can't recognize?
To ensure your survival, your brain evolved to avoid one thing: uncertainty. As neuroscientist Beau Lotto points out, if your ancestors wondered for too long whether that noise was a predator or not, you wouldn't be here right now. Our brains are geared to make fast assumptions, and questioning them in many cases quite literally equates to death. No wonder we're so hardwired for confirmation bias. No wonder we'd rather stick to the status quo than risk the uncertainty of a better political model, a fairer financial system, or a healthier relationship pattern. But here's the catch: as our brains evolved toward certainty, we simultaneously evolved away from creativity—that's no coincidence; creativity starts with a question, with uncertainty, not with a cut and dried answer. To be creative, we have to unlearn millions of years of evolution. Creativity asks us to do that which is hardest: to question our assumptions, to doubt what we believe to be true. That is the only way to see differently. And if you think creativity is a chaotic and wild force, think again, says Beau Lotto. It just looks that way from the outside. The brain cannot make great leaps, it can only move linearly through mental possibilities. When a creative person forges a connection between two things that are, to your mind, so far apart, that's a case of high-level logic. They have moved through steps that are invisible to you, perhaps because they are more open-minded and well-practiced in questioning their assumptions. Creativity, it seems, is another (highly sophisticated) form of logic. Beau Lotto is the author of Deviate: The Science of Seeing Differently.